The following appeared in DOW JONES NEWSWIRES October 8, 2003:
TO FINANCIAL ADVISORS
AND FUZZY MATH
BY KAJA WHITEHOUSE
Next time your financial adviser makes a prediction for an average rate of return during an investment pitch, you might want to doublecheck
the math.
Some financial advisers rely too heavily on a
formula known as arithmetic average, which can
be misleading when investing for the long term.
Financial advisers who use this formula may be
overstating your potential profit and leading
you to take risks you might otherwise avoid, academics and other financial professionals say.
Errors tend to widen when it comes to very volatile securities like emerging-markets stocks.
Arithmetic math involves a very simple formula,
which is probably why so many people rely on
it. To decide an average return, you add up all
the return percentages and divide the results by
the number of percentages.
It's a perfectly valid way to determine an average, as long as it's used to frame a stand-alone
one-year return, said Knut Larsen, a partner
with Brigus Group, a Toronto education service
for financial advisers.
The classic example to illustrate the flaws with
arithmetic math goes like this: You start with an
investment of $100 and it grows 100% the first
year and loses 50% the next year. To calculate
the total return using arithmetic math, you
would add the returns from both years -- in this
case 100 minus 50 -- and divide them by two, or
the number of returns.
That leaves you with the illusion of a 25% profit,
when in reality you're right back where you
started -- with $100. After rising 100% the first
year, you had $200; but a drop of 50% cut that in
half, back down to $100.
The alternative is known as geometric average,
or compound annual return. This takes compounding and volatility into consideration.
Unfortunately, geometric average is a complicated formula, involving cube roots, so it may
not be possible to figure out the results without
a spreadsheet. But the point is to educate yourself on the issue, not to memorize complex formulas, Mr. Larsen said. Simply understanding
when one formula should be used over the other, and knowing the flaws of arithmetic math is
a good start, he said.
When comparing the two results, the arithmetic
average generally ends up being higher than the
geometric average, said Campbell Harvey, a finance professor with Duke University's Fuqua
School of Business. For example, annual returns
on the S&P 500 index from 1927 until now are
about 12% using arithmetic math, and 10% using geometric math. That's a two percentage
point difference.
The deviation isn't always enough to get worked
up about, but it depends on factors such as volatility, and even fees and interest. For example,
the greater the volatility of the security in question, the greater the spread will be between the
two results, Mr. Harvey said.
He recalls feeling struck once by an advertisement touting Brazilian stocks attached to data
showing "incredible returns" of about 50% a
year. Knowing Brazil is a volatile market, Mr.
Harvey went back and applied geometric math
to the returns. His findings produced an average
return closer to zero.
Volatility can affect the portfolio in negative
ways because a severe drop makes it that much
harder to catch up on the reduced amount, even
if returns are phenomenal thereafter. But when
using arithmetic average, all that is known is the
one-year average return, not total results.
Misleading return projections using arithmetic
math are common in the insurance world, said
Peter Katt, an insurance analyst in Mattawan,
Mich. Some products require high return forecasts to make the products work, and this is one
way to get around that, he said, adding that
consumers need to educate themselves.
"I deal with very bright clients and advisers, and
they have no idea what I'm talking about" when
referring to the different formulas for calculating
results, he said.
It may seem like a lot of financial hocus-pocus,
but sometimes the misrepresentations aren't intentional, Mr. Larsen said. He published a primer on the subject this summer after bumping
into a financial adviser who legitimately didn't
know the effects arithmetic math was having on
his planning. The adviser had a client who suffered a portfolio loss of 45%, and the adviser believed the client would need an annual return of
15% a year to get back to the original investment
in three years. In reality, he would have to prepare for a return of more like 22% a year, according to Mr. Larsen's calculations.
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Tutorial: Geometric Mean in Excel
The data in column B are month-end values for the Standard & Poors 500 index for a 24month period. In column C these are converted into 23 months of percent changes (also
known as “returns”). A simple arithmetic mean is calculated in cell C27.
A
Month
Apr-02
May-02
Jun-02
Jul-02
Aug-02
Sep-02
Oct-02
Nov-02
Dec-02
Jan-03
Feb-03
Mar-03
Apr-03
May-03
Jun-03
Jul-03
Aug-03
Sep-03
Oct-03
Nov-03
Dec-03
Jan-04
Feb-04
Mar-04
B
C
S&P End % change
1067.14
989.82
-7.25%
911.62
-7.90%
916.07
0.49%
815.28 -11.00%
885.76
8.64%
936.31
5.71%
879.82
-6.03%
855.70
-2.74%
841.15
-1.70%
848.18
0.84%
916.92
8.10%
963.59
5.09%
974.50
1.13%
990.31
1.62%
1008.01
1.79%
995.97
-1.19%
1050.71
5.50%
1058.20
0.71%
1111.92
5.08%
1131.13
1.73%
1145.00
1.23%
1126.21
-1.64%
1107.30
-1.68%
D
E
F
G
1
2
=(B3-B2)/B2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
=AVERAGE(C3:C25)
25
26
=GEOMEAN(C3:C25)
27 Mean
0.2832% 0.1607% #NUM!
28
29
=B2*(1+C27)^COUNT(B3:B25)
=B2*(1+D27)^COUNT(C3:C25)
30
31
32 Predicted S&P
1138.85 1107.30
The difficulty is that this average will not result in a correct calculation of the ending
value of the S&P over 23 months, as shown in cell C32. Using the arithmetic mean of the
returns (0.2832%) results in an overly optimistic forecast (1138.85 instead of 1107.30).
There is an Excel function that helps calculate the geometric mean, but it won’t work if
any of the observations are negative (as is the case in this data set — see cell E27).
One way around this is to use the Excel Goal Seek feature. Goal Seek solves equations
with one unknown. In our case we want to find the unknown X, such that:
1067.14 * 1  X 23
 1107.30
Here’s how to set up Goal Seek for this problem:
Goal Seek comes up with a geometric mean of 0.1607%, which works correctly (see cell
D32).
368